Mathematical instrument



cs. J-DASHEFSKY 2,203,674

MATHEMATICAL INSTRUMENT Filed Dec. 28, 1938 5 Sheets-Sheet 1 June 11, 1940.

J3me 1940- G. J. DASHEFSKY LIATHEMATICAL INSTRUMENT Filed Dec. 28, 1958 5 Sheets-Sheet 2 J1me 1940- G. J. 'DASHEFSKY 2,203,674

MATHEMATICAL INSTRUMENT Filed Dec. 28, 1938 5 Sheets-Sheet 5 FIG. 12

' 1N ENTOR June 11, 1940. G. J. DASHEFSKY MATHEMATICAL INSTRUMENT Filed Dec. 28, 1938 5 Sheets-Sheet 4 FIG. l8

a V YR u lk Y m WMIAYU v1 Yb m N O I. m n A H w .0 w YA E1 8 m m 5 h t 31% w on 26 Pu P June 11', 1940. G H F Y 2,203,674

IATHEMATICAL INSTRUMENT Filed Dec. 28, 1938 5 Sheets-Sheet 5 Patented June 11, 1940 UNITED STATES MATHEMATICAL INSTRUMENT George J. Dashefsky,

Rockville Centre, N. Y.

Application December 28, 1938, Serial No. 248,040

8 Claims.

(Granted under the act of March 3, 1883, as amended April 30, 1928; 370 0. G. 757) This invention relates to an instrument for rapidly and accurately effecting the addition of vector quantities by mechanical means.

An object of the invention is to provide a device which reduces the addition of vectors to a mechanical process, requiring a minimum of skill and mental concentration.

A further object is to provide a simple instrument which is useful for the rapid evaluation of the resultants of a large number of vector diagrams such as are met in the study of properties of firing sequences of internal combustion engines, particularly as they affect their dynamic characteristics, such as unbalanced moments, torsional vibration, torque reaction, et cetera.

A further object is to provide an instrument, commonly known as a harmonic analyzer, for analyzing a periodic curve into its harmonic components, that is to obtain the mathematical coefficients of the Fourier Series frequently used in science and engineering.

A further object is to provide an instrument for summing up vector quantities met in the art of dynamic balancing of rotating machine members.

further object is to provide an instrument useful in the solution of certain problems dealing with alternating current electric circuits.

This invention should be of value in quickly and accurately solving any of the problems met in various fields, which either inherently are, or may be, reduced to a process of vector addition. Other objects of the invention will be apparent from the following description:

Figures 1 and 1(A) indicate the general arrangement of the instrument in one of its preferred forms.

Figure 2 shows a second form, particularly as regards the means for supporting the disk I.

Figure 3 shows a typical vector diagram and its resultant obtained by graphical closure of the vector polygon.

Figures 4 and 5 show a second method of summing up the same vectors shown in Figure 3, by

resolving the original vectors into their co-ordinate components.

Figures 6, '7, and 8, showing the instrument in elemental form, illustrate the underlying mathematical principle.

Figure9is a diagrammatic representation of the crankshaft of a 5-cylinder, 2-cycle engine, used to illustrate an application of the invention to the problem of engine balance.

Figure 10 is the familiar vector diagram for primary moments for the crankshaft of Figure 9.

Figure 11 indicates the arrangement of weights on the disk of the instrument, corresponding to the vector diagram Figure 10.

Figure 12 is the vector diagram for secondary moments for the crankshaft of Figure 9.

Figure 13 is the arrangement of weights on the disk of the instrument, corresponding to the vector diagram, Figure 12.

Figure 14 represents a typical relative amplitude curve of torsional vibrations, for an installation comprising a 'l-cylinder, 2-cycle reciprocating engine and one principal revolving mass. This representaive installation and curve are used to illustrate an application of the invention to the solution of problems in synchronous torsional vibration.

Figure 15 represents a vector diagram, whose resultant, 2,6 is a measure of the exciting torque causing torsional vibrations. As represented, 23 is for the 3rd harmonic of the turning eflort.

Figure 16 shows the arrangement of the instrument disk and weights corresponding to the vector diagram of Figure 15.

Figure 17 represents a periodic curve to be analyzed into harmonic components, and used in illustrating application of the invention as a harmonic analyzer.

Figure 18 is a curve derived from Figure 17 in a manner later described, and is used in describing the mathematical principle of the instrument when used as a harmonic analyzer.

Figure 19 represents the arrangement of the instrument disk and weights to determine, as an example, the first harmonic coefficients.

Figure 20 represents the arrangement of the instrument disk and weights for obtaining the 3rd harmonic coefllcients as a further example of the use of the invention as a harmonic analyzer.

Figs. 21 and 22 are plan and side elevational views, respectively, of an alternative means, similar to that of the pendular system shown in Fig. 1, for providing an equilibrating force to act on the disk.

Figures 23 and 24 indicate a basic form of the instrument to further illustrate the various mechanical arrangements which may be used to embody the purposes of the present invention.

One embodiment of the invention is shown in Figures 1 and 1(A). One of several interchangeable disks 1 is carried by member 2, which is mounted in gimbals in the manner of 3. nautical compass. Member 2 is free to rotate in a pair of pivot bearings 3 carried on the ring 4. The ring 4 in turn is also mounted in pivot bearings I, carriedby stanchion: 8, permitting rotation about an axis at right angles-to the axis of rotation of the disk I. The pivots are arranged so that the mutually perpendicular axes of rotation pass through the center of the disk I. The ring 4 is supported on a base I, which may be levelled oil by means of screws 8. Member 2 carries a light projection system comprising a lamp 8 with a filament of the concentrated type. a lense I8, carried by tube II, which telescopes into tube I2, residing in member 2. The disks I are retained on 2 by means of nut I2A. Current is fed to the lamp by very flexible leads passing through holes in 2. An image of the lamp filament is focused on a translucent screen I3 graduated in polar co-ordinates. The graduated screen, sandwiched between two plates of glass, or plastic I4, is supported from the base by stanchions I5 and secured thereto by nuts Ii, may readily be replaced with other screens graduated in any required manner to permit direct readings in units pertinent to the problem in hand. Deflections of the disk I are measured by the displacement of the light spot from the center 01 the screen I8 indicating both magnitude and direction.

A threaded rod Il, forming the lower part of 2, carries a movable weight I8, by means of which the instrument sensitivity and scale reading is controlled.

The disks I' are provided with pegs 22 with appropriate angular spacing. Weight holders 28 carrying preweighed groups 0! washers 24 of 1 unit, 0.1 unit, and 0.01 unit may be shifted from peg o peg as requisite. The weights 24 representing the vector magnitudes are thus adjustable by a simple counting process rather than by weighing.

The oscillations of the disk in its gimbals are damped out by vanes I8, carried at the extremity of rod I1, and moving in a dish 2|, containing a suitable damping fluid 20. The time required for the disk to come to rest is thus greatly reduced.

Referring again to Figure 1, it will be noted that the disk I is free to rotate about each of the mutually perpendicular axes. It may of course rotate simultaneously about both axes. Any tilting action of the disk due to weights 24 is balanced by the pendular weight I8, which assumes a deflected position about the two axes of rotation, in accordance with the magnitudes of weights 24 and their disposition on the disk I. Actually the weight of the light projection system, 8, I0, II, and I2, member 2, and threaded rod I1, must also be considered. However, there is a net weight, W, which represents the combined eflect of all of these. See Figures 6, 7, and 8.

The deflection of the disk I, in its gimbals, is equivalent to supporting it on a single eflective pivot 25 carried on 28, as shown in Figure 2. Such an arrangement could be used in a second form of the instrument, and would possess all of the mathematical characteristics of the instrument shown in Figure 1.

Refer to Figure 3 representing the addition of six vectors. The resultant vector is obtained by the usual graphical construction of the vector polygon in which line OP represents the vector sum or resultant.

A second method of adding these vectors is shown in Figures 4 and 5. Each vector is resolved into its co-ordinate components as shown in Figure 4, the respective components being added in the manner indicated in Figure 5. The two resultant components Eh and Ev may then be combined to give the resultant vector.

The following example involving the addition 01' vectors shown in Figure3 will demonstrate the mathematical principle of the instrument. Refer to Figures 6, 7, and 8, showing the instru ment in elemental form; Each vector is represented by a weight placed on the disk at a common radius, 1. These weights have the same relative values as the original vector magnitudes, while the direction of the vectors is cared for by disposing the weights on the disk in the same angular relationship as the original vectors. The weights so disposed result in a net moment about axes X-X and YY, producing rotation about these mutually perpendicular axes, displacing the net pendular weight W from its previously plumb position. The net moments of weights 101 to we inclusive, about the two axes, are balanced by the moment of the displaced weight W. For this condition of equilibrium,

About YY:

In general for any number of weights and orientation,

WR sin on=r (cos 4m) 2(w sin 0) About XX:

WR sin v=(wir cos (ii-Himcos 02+!!! cos 01+ um cos 04+wur cos os+wur cos 0:) cos v (Equa. 3)

In general for any number of weights and orientation,

WR sin v=r (cos v)2(w cos 0) (Equa. 4)

The optical system 9 to I I, attached to the disk I projects a spot of light P, on to the graduated screen II. The co-ordinate displacements of this point of light from the center of the screen as indicated in Figures 6, 7, and 8, are:

(Equa. 2)

Therefore substituting Equations '7 and 8 into Equations 2 and 4, it follows that:

D EUV sin 6) (Equa. 9)

D %E(W cos 0) (Equa. 10)

It is apparent from Equations 9 and 10, that DH and Dv are measures of the co-ordinate components of the vector sum of the six original vectors shown in Figure 3, and represented by equivalent weights w to we inclusive, as shown in Figure 6. Comparing Figure 8 with Figures 4 and 5, it may be noted that the lines Da and Dv in Figure 8 correspond to 2h and 2v in Figure 5, while GP in Figure 8 corresponds to the resultant vector OP in Figure 5. The direction of the resultant is also indicated directly on the screen I8.

Examining Equations 9 and 10, it may be noted that the scale or sensitivity of the instrument may be controlled by varying WR. Moving the pendular weight I8 toward the disk, decreases R and magnifies the readings. This offers a convenient means for calibrating the scale in terms 0! the pendular weight until a scale reading 1.05

obtains. Subsequent readings of vector sums are therefore readily obtained by multiplying scale readings by 100.

The actual size of the weights w and w: representing the vectors is not important, it is only necessary that they be in the same proportion as the original vectors.

The condition of engine balance is intimately associated with the choice of firing order. It is beyond the scope of this application to discuss this problem except as it is necessary to demonstrate application of the instrument to such problems.

Figure 9 shows the crankshaft of a -cylinder. 2-cycle engine taken as an illustrative example. Due to the reciprocating weight and crankshaft angularity there exists at each crank inertia forces acting in the line of piston stroke. There is no horizontal component.

The inertia force at each crank is expressed by:

2 2 Fl= %f%%(cos 9+? cos 26+ (Equa. 11)

This indicates that the inertia force is equivalent in effect, to a series of harmonic forces recurring once-per revolution, twice per revolution, etc. The first two harmonics are the most important, but the 4th, 6th, etc. harmonics, though progressively smaller, also exist. In very .high speed engines the 4th harmonic may be of importance. Limiting discussion to the first two harmonics, the presence of two separate forces may be considered.

Where:

F1=Primary force-lbs. Fn secondary force--lbs.

N=Engine R. P. M.

Z=Length connecting rodinches rc=Crank radios-inches g=Gravitational constantinches/sec./sec.

0=Angular displacement of crank from T. D. C.

of cylinder No. 1 taken as reference position.

Take cranknumber l as a reference point for expressing moments. The'same force exists at each cylinder. The moment of any cylinder is proportional to its distance from the reference point. The phase relation for successively firing cylinders is 72 for the primary forces and moments and 2x72=l44 for secondary forces and moments. Assuming equal cylinder spacing, the moments of cranks 2, 3, 4, and 5, are in the proportion of 1:2:3z4. The moment of crank numher 1 is zero. Vector diagrams for primary and secondary moments are shown in Figures and Consider a disk I, shown in Figure 11, having 5 pegs, disposed 72 degrees apart, and weights in the proportion of 0, 1, 2, 3, and 4 selected to represent respectively, the moments of the cranks 1, 2, 3, 4, and 5. Each of these weight groups is marked with the cylinder number. To evaluate the vector diagrams the following procedure is followed. Temporarily place one of the weights, preferably 102 which represents one weight unit, on the peg at 0:0 that is, the top peg shown in Figure 11. The 'pendular weight I8 Figure 1, is now adjusted until a convenient scale reading is obtained in terms of the primary moment:

41 mm m (Equa.

Having thus calibrated the instrument, place the weights on the pegs in proper phase relation with respect to firing order. There is no weight for w; since crank No. 1 has zero moment. For the primary moment the vectors are 72 degrees (one interval on the disk) apart and the weights are disposed as shown in Figure 11. The vector sum and the direction of the resultant are indicated directly on the screen in the manner shown in Figure 8.

To obtain the resultant secondary moment, that is, to evaluate the vector diagram, Figure 12, readjust the scale, in the manner described for primary moments. Since the phase relation is 2x72 degrees for secondary moments, dispose the weights, in firing sequence, on every other peg as shown in Figure 13.

In similar manner, the 4th harmonic may be obtained by putting these same weights on every fourth peg. Higher harmonics are similarly summed up by distributing the weights counting off each time a number of peg spaces equal to the harmonic under consideration.

The moment of any crank is determined only by its distance from crank No. l, and is independent of firing order. The resultant however is influenced by firing order. Once having calibrated the instrument and fixed the weights representing cylinder moments, resultant unbalanced moments for a large number of firing orders may be quickly investigated.

For the 5-cylinder, 2-cycle engine the number of firing orders possible is,

(nl) 4 3 2 1 f ""2 By systematic shifting of the weights, a reading for each of the possible firing orders is quickly obtained. A similar procedure is now carried out for the secondary moments, remembering that in reading off the firing order, successively firing cylinders are displaced by two peg spaces, or 144 degrees. If required, similar procedures may be carried out for the higher harmonics.

Thus far, moments due only to the inertia of reciprocating weights have been considered. These inertia forces act vertically only, having no horizontal component. Centrifugal forces due to unbalanced weights also produce moments of importance, having components in both vertical and factorial l2 horizontal planes of the engine. They are ofprimary frequency only. Higher harmonics do not exist. The response of the engine and structure in the two planes is usually different, and

it is desirable and customary to determine the a ment readings permits choice of a suitable firing order.

Disks may be arranged to suit any particular configuration of cranks and cylinders, such as V engines, radial, opposed piston engines, etc.

The instrument is useful in solving a certain phase of the torsional vibration problem. Reviewing the problem briefly:

The input of vibration energy (inch pounds) in a multi-cylinder engine is:

=m WK B) (Equa. 15) Where,

a=Amplitude of vibration at the free end of the crankshaft-degrees.

Mv fiarrnonic torque of cylinder for particular order of vibration under considerationpound inches.

Jfi=Vector sum of the amplitudes of vibration at the respective cylinders, their relative phase being that of the successively firing cylinders.

The phase relation for a given order of vibration =m0 (Equa. 16)

Where,

'y=Phase angle-degrees. m=0rder of vibration (number of vibrations per revolution of engine). 0=Crank displacement between successively firing cylinders.

In a critical speed a balance exists between the input energy, Kr, and the damping energy, Kn. Assume the following law of damping, where a is the amplitude of vibration, in radians,

a TsOMytflE S) =K a i Mvo ii 180 K It is apparent that the amplitude of vibration depends on 2,9. Examining Equation 16 as applied to the 'l-cylinder 4-cycle engine of Figure 14,

a=)\1 (Equa. 17)

720 '7 degrees Table I 'y-ile Orders oi 6% 1, 6 5 4 Vibration (in) 7a,, 13% 8, l3 8%, 12% 9, l2 9%, 11% 10, ll

Phase angle in terms oinumber of peg intervals](eag(l)1 1 2 3 4 5 6 interva =7 1n) In Figure 15 is shown a typical vector diagrani for the third order vibration associated with the relative amplitude curve of Figure 14, and the firing order 1235764. To evaluate this vector diagram with the instrument, in the now familiar manner, weights in proportion to the p values are placed on the seven peg disk, in proper phase relation. The screen is calibrated by placing the weight representing ,8 on the disk and adjusting the pendular weight II to a position giving a suitable scale.

For each iiring order, the whole range of orders may be investigated by simply shifting the weights to their appropriate angular positions. Starting with the one order, each weight, in sequence of firing, is placed on the pegs in succession, progressing in a clockwise direction. For the second order, the weights are placed on every second peg; for the third order, every third peg, and so on. Usual interest would center about the magnitude of the resultant, the phase being unimportant, so that in the 'l-cylinder engine there are only 3 groups of minor orders to be evaluated.

In any particular problem, the p values are fixed, and a change in firing order does not call for readjustment of the weights representing [3 values. Thus to investigate the whole range of possible firing orders calls merely for a proper shifting of these weights. The values of 25 for a whole set of firing orders may thus be quickly and accurately determined. In the 'l-cylinder engine there are 360 possible firing orders. Since there are 3 groups of minors, the complete investigation involves 1080 shifts of weights. With the aid of the present invention this appears to be an entirely feasible task requiring a minimum of skill and care.

The instrument may be used as a harmonic analyzer. Any periodic function such as the curve illustrated in Figure 17 may be represented by Fouriers trigonometric series, so that,

T:f(0)-=Ao+A1 sin 8+Az sin 20+A3 sin 30+ A, sin N0+B1 cos 0+8: cos 20+B3 cos 30+ B cos N0 (Equa. 18)

The coeflicient, A0, represents the mean value of the function while the coeiiicients A1, A2, etc. and Bi, B2, etc. represent respectively the sine and cosine harmonic components of the curve.

These are important coefiicients in many engineering problems and it frequently becomes necessary to evaluate them. Schematic tabulations and mechanical devices are available for performing harmonic analyses. The present device is suited to a speedy determination of one or many of these coeilicients by a simple mechanical procedure requiring a minimum of care and concentration.

Representing the periodic function as occurring over 21r radians it can be shown that,

2 A 0 'mo (Equa. 19

1 A,,,=; o T.s1n m9.d6 (Equa. 20)

Am, Bm, represent the harmonic coefiicients for the mth harmonic of the Fourier series of Equation 18.

The coeflicient A0, is obtained from evaluation of Equation 19, where the integral represents the net area under the original, curve. The integral involved in the expression for Am is the area unoriginal function by the appropriate sin (me).

A graph of t==T.sin m0 is shown on Figure 18. The area under this curve multiplied by is the value of Am. The values of Bm may be obtained in a similar manner, excepting of course,

that the auxiliary curve t=T.cos m0 is obtained by multiplying ordinates of the original curve by the appropriate cosines of me).

As a general matter, the process for determining the areas representing the intervals of Equations 19, 20, and 21, is not important. However, in adapting the present instrument as a harmonic analyzer, the following method of obtaining areas is desirable.

The abscissa of the original function is divided so as to give 1' equally spaced ordinates at (f-l) equal intervals of A0. The ordinates at each point are multiplied by the appropriate sine or cosine value giving a new series of ordinates on the curves,'t=T.sin m0, and t=T.cos m0. Refer to Figure 18. The first elementary area or assumed as a trapezoid may be closely approximated by,

Adding up all of the elementary areas, a1, 02, as, etc. gives the net area of the curve, representing the value of the integrals of Equations 20 and 21.

The total area under the curve is therefore:

For the purposes of illustration, the curve is to be analyzed into its harmonic components by using 24 ordinates. A disk with 23 pegs such as shown in Figures 18 and 19is therefore to be used. Weights corresponding to the 24 equally spaced ordinates on the curve are prepared, the first and last ordinates being represented by weights corresponding to half of these ordinates, ordinates being measured from line BB, Figure 17. Weights should be suitably marked for ordinate identification. The instrument is adjusted in the manner previously described, to give a suitable scale. A0 is-merely the algebraic sum of the ordinates measured from A-A, Figure 17, multiplied by v and the use of the harmonic analyzer will not' ing the weights, placing 101 on the first pe w:

Thus any harmonic, say the mth one, may be obtained by distributing the weights in succession,

counting ofi. m peg intervals each time. Figures 19 and 20 show, as examples, the distribution of weights for the 1st and 3rd harmonics of the curve of Figure 17, for a 24-ordinate analysis.

Greater accuracy will, of course, attend the use of larger number of ordinates, but the use of pose of illustrating the principle and use of the invention. The instrument is adaptable to many fields of engineering.

Several groups of weights, each group on a concentric circle, may be used to evaluate families of vectors.

The instrument may also be used to resolve a vector into components in any desired directions. The vector to be resolved, represented by a weight, is placed on one of the pegs. On the other pegs representing the direction of the component vectors, (but shifted through 180- on the disk) are placed weights until the disk returns to its undeflected position. The weights required to balance the disk represent the magnitudes of the component vectors and the direction diametricai 1y opposite these weights are the directions of the component vectors.

Likewise the construction of the instrument should not be limited to the forms disclosed in this specification. The light projection system need not be used, a simple pointer extending from the disk could be used to indicate deflections. Also the light system could be so arranged that the projector is a fixed element, while the disk carries a mirror at its center, which reflects the beam of light upon a, screen in the same manner as used in certain types of galvanometer. Again, instead of a pendular mass attached to the disk, a movable weight could be arranged, adjustable both as to magnitude and angular position. This weight could be varied until the unbalanced moment of the disk were counterbalanced and the disk brought back to its initial position in a manner analogous to that employed in the ordinary beam scale. Or the unbalanced force could actually be measured by direct-reading weighing scales.

An optional system oi equilibrating forces is shown in Figures 21 and 22. Three springs 21, of the same load-deflection characteristic connect the rod I1 and lugs 28 secured to the base I. All springs are under the same initial tension. The manner of loading the disk is not limited to the system of weights at constant radius used in the forms disclosed. Fixed weights may be used and the radius varied. It is only necessary that the moment of loads on the disk, representing a system of vectors, be such that the moments oi these loads about the center of the disk are in the same proportion as the vectors. Therefore hydraulic, magnetic, or any desired form of loading may be used.

As a further example of the diverse ways in which the purposes of the present invention may be accomplished, reference is made to Figure 24. This embodiment comprises a hemisphere 3|, terminating in a plane flange 32, which serves as the disk I of Figure 1. As previously described, the disk is loaded by weights 24, carried on appropriately located pegs 22. The hemisphere 3i and flange 32 are made of relatively light material and provided with a heavy inset 33. The concave spherical surface is graduated into meridians and lines of latitude as indicated. The sphere is free to roll on a smooth lever surface 3. Under action of the weights 24, the disk will roll to a position of equilibrium between the displaced center of gravity of the sphere and flange and the weights. This displacement is a measure of the unbalanced moment on the disk, and is consequently, as previously discussed, a measure of the resultant vector sum of the vectors represented by the weights 2|. The displacement of the sphere is indicated by the position of the ball 35, which remains always at the lowest point of the concave surface.

The description of the invention has been restricted to the embodiments shown in the specification and to illustrative uses described therein. The scope of the invention is not limited to ,iese embodiments and uses, but only to the scope of the following claims.

The invention described herein may be manufactured and/or used by and for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or thereafter,

Having thus set forth and disclosed the nature of this invention, what is claimed is:

1. An instrument for mechanically obtaining the resultant of a plurality of quantities having the properties of magnitude and directional relationship with respect to a common reference point: comprising a disk pivoted in effect at its polar center, means for loading said disk in proportion to the magnitudes of said quantities and in the same angular relationship as obtains there-among, and means for measuring the unbalanced moment on said disk as an index of said resultant quantity.

2. An instrument for mechanically obtaining the resultant of a plurality of quantities having the properties of magnitude and directional relationship with respect to a common reference the resultant of a plurality oi quantities having the properties of magnitude and directional relationship with respect to a common reference point; comprising a disk pivoted in effect at its polar center; means for providing a counterbalancing force on said disk, weights representing the magnitudes of said quantities, means for retaining the weights on said disk in the same angular relationship as that of said quantities, and means for indicating the deflection oi said disk, the deflection being a measure of the magnitude and direction 0! said resultant quantity 4. An instrument for mechanically obtaining the resultant of a plurality of quantities having the properties of magnitude and directional relationship with respect to a common reference point; comprising a disk pivoted in eflect at its polar center, a pendular weight attached to said disk, weights in the same proportion as the magnitudes of said quantities, means for retaining said weights in the same angular relationship as exists among said quantities and means for indicating the deflection of said disk as a measure of the magnitude and direction of said resultant quantity.

5. An instrument for mechanically obtaining the resultant of a plurality of quantities having the properties of magnitude and directional re lationship with respect to a common reference point; comprising a disk pivoted in effect at its polar center, a pendular weight attached to said disk, weights representing the magnitudes 01' said quantities, means for retaining said weights in the same angular relationship as that existing among said quantities, a screen, and means for projecting a spot 0! light on said screen for indicating deflections of said disk, said deflections representing the magnitude and direction of said resultant quantity.

6. An instrument for mechanically obtaining the resultant of a plurality of quantities having the properties of magnitude and directional relationship with respect to a common reference point; comprising a disk mounted in gimbals, a pendular weight attached to said disk, weights representing the magnitudes of said quantities, means for retaining said weights on said disk in the same angular relationship as exists among said quantities, and means for indicating the deflection of said disk as a measure of the magnitude and direction of said resultant quantity.

7. An instrument for mechanically obtaining the resultant of a plurality of quantities having the properties of magnitude and directional relationship with respect to a common reference point; comprising a disk mounted in gimbals, a pendular weight attached to said disk, weights representing the magnitudes of said quantities, means for retaining said weights on said disk in the same angular relationship as that which exists among said quantities, a screen and means movable with said disk for projecting a spot oi 10 means for retaining said weights on said disk in the same angular relationship as that which exists among said quantities, a screen, means mov-.

able with said disk for projecting a spot of light on said screen as a measure oithe magnitude and direction oi said resultant quantity, and meansi'or adjusting the distance of said pendular weight from said disk, thereby varying the sensitivity and scale 01 said instrument.

GEORGE DABHEFSKY. 

